We plot in Figs. REF (a) and REF (b) the dynamic evolutions of the populations \(P_{b,3}\)  and \(P_{a,4}\) , respectively, where \(P_{j,3(4)}\)  (\(j=a,\,b\) ) denotes the population \(|c_{j}|^{2}\)  of emitter \(j\)  with the initial state \(|\psi _{3}(0)\rangle =|e,g,g,0\rangle \)  (\(|\psi _{4}(0)\rangle =|g,e,g,0\rangle \) ). We find that in the giant-atom trimer, nonreciprocal excitation transfer emerges again for \(\theta \ne m\pi \)  due to the asymmetric overall coupling between \(a\)  and \(c\) . Different from the behaviors in Figs. REF  and REF , \(P_{b,3}\)  and \(P_{a,4}\)  oscillate here with a fixed phase difference. To understand this difference, we also plot in Figs. REF (b) and REF (c) the evolutions of the populations of all three emitters in the case of \(\theta =\pi /2\) , with initial states \(|\psi _{3}(0)\rangle \)  and \(|\psi _{4}(0)\rangle \) , respectively. One can find that for both initial states, the excitation exhibits a directional circulation along the same direction of \(a\rightarrow c\rightarrow b\rightarrow a\)  after \(t=d/v_{g}\) , which is a signature of broken time-reversal symmetry that cannot be observed for \(\theta =m\pi \) . That is to say, the single excitation initially prepared in emitter \(b\)  is preferentially transferred to \(a\)  while that initially prepared in emitter \(a\)  shows a preferential transfer to \(c\) . Note that circulations along the opposite direction \(a\rightarrow b\rightarrow c\rightarrow a\)  can be achieved for \(\theta =-\pi /2\) . This is analogous to chiral currents of electrons in an Aharonov-Bohm cage and of photons in a synthetic magnetic field [1]}, [2]}, [3]}, [4]}. In this way, the target emitter (\(b\)  for \(|\psi _{3}(0)\rangle \)  and \(a\)  for \(|\psi _{4}(0)\rangle \) ) can be excited efficiently within different durations for the two initial states. Note that another initial state \(|\psi _{5}(0)\rangle =|g,g,e,0\rangle \)  results in essentially the same dynamics as those with \(|\psi _{3}(0)\rangle \) . This is because Eq. (REF ) is invariant by exchanging the amplitudes \(c_{a}\)  and \(c_{c}\)  and reversing the coupling phase \(\theta \)  (with identical \(\theta \) , \(|\psi _{3}(0)\rangle \)  and \(|\psi _{5}(0)\rangle \)  lead to exactly reverse circulations). Moreover, since emitter \(b\)  is not requested to interact directly with \(a\)  or \(c\)  in this case, the trimer model can be used for nonreciprocal excitation transfer between far apart emitters, which is of potential applications in large-scale quantum networks.
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